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When is a goal not a goal?

So what's the problem?

Graphic 1: The definition of a goal.

The definition of a goal has been clear ever
since football codified its rules in the mid-19th century: for a goal to be scored, all of
the ball must cross all of the line.
So in the diagram on the right, C is a goal whereas A and B
are not. Most disputed goals result from
defenders illegally returning the ball into
play after it has crossed the line, but
significant numbers occur without any
human intervention at all. Perhaps the most
famous example is Geoff Hurst's second goal
at the 1966 World Cup final between England
and West Germany. His shot cannoned off
the crossbar and was judged to have
bounced behind the goal line before
rebounding back into play.

There have been
other controversial examples over the years. After Hurst's, the highest-profile incident
occurred at the 2010 World Cup in the
second round match between England and
Germany (again!), when Frank Lampard's
shot struck the crossbar and bounced a good
half-metre behind the line. On this occasion the goal was disallowed by the match
officials, but hundreds of millions of
viewers around the world saw it
repeated in high-definition, slow
motion replays.

Why such incidents occur is perplexing
at first sight. In the diagram below the ball hits
the bar and because the ball is
travelling towards the rear of the goal,
surely it will continue in that direction
after striking the ground? This ignores
the considerable backspin the ball
acquires in its contact with the bar.

If this spin is great enough the ball's
horizontal velocity can be reversed
when it bounces on the ground and if it
doesn't bounce too far behind the line
it can clear the crossbar in the recoil,
and bounce out. This event is so
fleeting, only two tenths of a second
from bar to ground, that it's hardly
surprising that referees make snap
decisions which are often incorrect. (You can find out more about ball spin in the Plus article If you can't bend it, model it.)

Graphic 2: Spin reverses the ball's direction.

We set out at the University of Bath to
investigate this behaviour. Can a ball be
made to reverse its velocity under
experimental conditions and what are the important factors that
govern
this behaviour?

Machines are more
consistent than footballers

For our study we used a programmable ball launcher. This was
absolutely essential because no professional, not
even one with the skills of a David
Beckham, could repeatedly fire shots at
an experimental crossbar with
sufficient accuracy to ensure that all of
the critical events would occur. The photo below shows how our experiment was set up.

Graphic 3: The ball launcher.

To film the impacts a high-speed digital
camera was situated to the side of the experiment so that it filmed the path of the ball as it travelled towards the crossbar. Over 50 impact events
covering a range of speeds and launch
angles were recorded. The photos below show two examples in freeze frame
format. One is clearly a goal, the ball
bouncing too far behind the goal line to
spin back. The other shows the ball's direction reversing with the ball rebounding back
into play. Just how we turned these
images into data files where accurate
positional coordinates, ball velocities
and spin rates could be obtained is
explained in the section which follows.

Graphic 4: This is a goal.

Graphic 5: And this is not.

From digital images to real-world coordinates: the DLT method

High-speed video can be enormously
helpful in developing sporting
excellence in a coaching context
but it is practically useless if
quantitative measures are required.
Such questions as "how fast is that
athlete running" or "what forces are
acting on the ball" can only be
answered if the movement field
is in some sense calibrated. This is
sometimes done by introducing devices
such as measuring rods in
the digital camera's view, but there is
no guarantee that the camera
reproduces its images linearly
right across the field and so a more
sophisticated technique is needed for
accurate work.

Let’s suppose a ball is moving with real-world coordinates given in metres relative to a fixed origin . A camera will record an image which can later be analysed on a computer screen. Suppose the ball’s position in this virtual world is where and are conventionally measured in pixels. By looking at a mapping of the real-world coordinates through the camera lens onto the screen it can be shown that and are related by the following equations:

This mapping is known as the direct linear transformation (DLT) and the parameters up to as the DLT coefficients. If these numbers were known, for example if they were absolute constants for the particular camera, then by measuring and and knowing up to it would be possible to back-calculate for and using the above equations. Sadly, life’s not that simple. The DLT parameters are specific to the camera’s position, how much it has been zoomed, the aperture chosen and so on. So they need to be determined for each particular case.

The way this is done is to place a calibration object with known coordinates in the field of view. Its apparent position on the screen can then be measured in pixels. We now have two equations in eight unknowns and very quickly see that adding another calibration point would give a further two equations. In fact four calibration points do the trick; eight independent equations, eight unknowns so the problem is solved and up to are known for that particular experiment.

Of course, we don't stop there.
A minimum of four calibration points
would not produce very accurate
results and the aim is to spread the
calibration throughout the entire
movement field. This can be done in
a variety of ways but one common
method is to use a vertical pole with
marker points placed very accurately
along its length (see the photo on the left). Then
by stepping the pole through
a series of known positions and
photographing these using the camera
we can build up a calibration grid. This
introduces lots of redundancy into the
problem. A hundred calibration points
would generate 200 equations from
which all different combinations of
eight equations in the above format can
be chosen and solved. Simple
averaging then gives accurate values
of the DLT coefficients.

The computation is quite intensive, but a mathematical software tool, such as
Mathcad, can
handle simultaneous equations and makes
very short work of the process. All that
remains is to film a sequence of ball
impacts against the crossbar and the
ground. For these, the ball is
appropriately marked so that its spin
can be tracked throughout the flight.
There follows a long, tedious sequence
of digitising, during which the will to
live can occasionally be questioned.
However, long careful digitising
sessions are good for the souls of
aspiring engineers and (occasionally)
their supervisors.

Finally, having determined the true
coordinates of the ball as described
above, trajectories can be fitted to the
data. Pre- and post-impact
velocities together with the
ball's angular velocity quickly follow. The diagram below shows an extract
from the processed data where the
ball's position is tracked through a
short sequence illustrating a goal (the green path), a
bounce where the forward velocity is
arrested (the blue path) and finally one where the
horizontal velocity is reversed (in red).

Graphic 7: Extract from data showing measured trajectories.

What do these experiments
tell us?

The results confirm the importance of
spin acquired during the ball's impact
with the crossbar and show that under
the right conditions a ball can easily
cross the line and rebound back into
play, to the consternation of both the
players and the match officials. In our
work, the critical zone – the region
behind the goal line from which a
spinning ball would bounce out – was
very narrow, about 35cm, not much
greater than the ball's diameter of
22cm. This was because our surface
was very hard, leading to a mean
coefficient of restitution (a number
measuring the elasticity of the impact)
of 0.75. It is generally accepted that
natural playing surfaces have much
reduced coefficients compared with synthetics, typically as low as 50% of
the synthetic value for really soggy turf.
So we set out to use our data to model
crossbar impacts with coefficients typical
of natural surfaces.

In doing this we used a simple model of
the ball's impact with the surface. Both
were assumed to be rigid, the elasticity
of the bounce being modelled by using
appropriate coefficients. By considering
the linear and angular momenta before
and after impact it is possible to
formulate simple equations relating
pre- and post-impact velocities and
spins. For the horizontal and vertical
velocity components we have:

(1)

(2)

In these equations, and are the pre- and post-impact velocities. The suffixes and denote the horizontal and vertical components. The coefficient of restitution is is the ball’s radius and the spin acquired during the ball’s impact with the crossbar. A glance at (1) shows that the horizontal velocity will be reversed if because if this is the case the velocity changes sign. A quick calculation confirmed that this was so for certain values of these quantities in the experiments.

Typical spin rates and recoil velocities
following the ball's impact with the
crossbar were taken from our
measurements: these were 75 radians per second
and 17 metres per second respectively. A coefficient of restitution of
0.45 (60% of our measured values) was
used to reflect the generally softer
playing surfaces encountered in
practice. The ball was assumed to be
projected downwards at a range of
angles following impact with the bar.
The post-bounce conditions could then
be determined from (1) and (2) and the
ball's trajectory after the bounce
determined assuming that it
subsequently moved under gravity alone.

The diagram below shows the results. The ball's
behaviour has been modelled from the
point where it stops moving horizontally (the red trajectory) to the
point where the reversal of the ball's motion would
lead to a second impact with the
crossbar as the ball bounced out.
Impacts in the green zone would lead to
a conventional goal; the ball's velocity is
modified but not reversed. Within the
blue zone the ball reverses but the
ground impact positions are too far from
the goal line to enable the ball to escape
by clearing the crossbar. Red is the
danger area. Under the assumed
conditions, balls bouncing here would
be returned into play. The coefficients used
(60% of our experimental findings) show
that the critical impact zone extends to
54cm behind the goal line. This is in very
good agreement with Frank Lampard's
2010 World Cup goal described earlier.

Graphic 8: Modelled impacts and progression to bounce-out.

The case for goal line
technology

Following the farrago of England's nongoal
at the 2010 World Cup, FIFA, the
organising body for international
football, seems to have had a change of
heart concerning the role technology
might play in this important area.
At the time of writing two possible
solutions are under intensive
investigation: England's Hawk-Eye
system relies on visual tracking of the
ball and its rival, the Danish product
GoalRef, on an electronic device in the
ball's interior. Both must report the
ball's position relative to the goal line
with high precision and it seems that a
choice will be made by FIFA in Kiev,
immediately following the final game of
the Euro 2012 tournament. The most
likely timing for its introduction would
then seem to be the 2013-14 season for
the English Premier League; but given
that Major League Soccer starts earlier
in 2013, the United States may well
pioneer the system. Watch out America!

Of course if either system does its job it
doesn't matter whether the goal resulted
from a crossbar impact or not.
But suppose the evaluations are
inconclusive. Then it might be profitable
to think about mitigation measures to
eliminate crossbar impacts as an
important source of disputed goals.
This could be done by making crossbar
impacts "spin neutral" and we at Bath
have some ideas on the matter.
Let's see what happens.

About the author

Ken Bray is a theoretical physicist and a Senior Visiting Fellow in the Faculty of Engineering and Design at the University
of Bath, UK. He has made a special study of the mathematics of football and the factors affecting the ball's flight, and has
lectured, broadcast and published widely for both academic and general audiences. He has gone on record as saying that as much as 30% of a footballer's technique is down to an intuitive
understanding of maths and science (although they shouldn't go anywhere near a computer!).